1 / 43

Testing multifactor models

Testing multifactor models. Plan. Up to now: Testing CAPM Single pre-specified factor Today: Testing multifactor models The factors are unspecified!. Detailed plan. Theoretical base for the multifactor models: APT and ICAPM Testing when factors are traded portfolios Statistical factors

naida-alvarado
Download Presentation

Testing multifactor models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Testing multifactor models

  2. Plan • Up to now: • Testing CAPM • Single pre-specified factor • Today: • Testing multifactor models • The factors are unspecified! EFM 2006/7

  3. Detailed plan • Theoretical base for the multifactor models: APT and ICAPM • Testing when factors are traded portfolios • Statistical factors • Macroeconomic factors • Chen, Roll, and Ross (1986) • Fundamental factors • Fama and French (1993) EFM 2006/7

  4. APT • K-factor return-generating model for N assets: Rt = a + Bft + εt, • where errors have zero expectation and are orthogonal to factors • B is NxK matrix of factor loadings • Cross-sectional equation for risk premiums: E[R] = λ0l + BλK • where λ is Kx1 vector of factor risk premiums • ICAPM: another interpretation of factors • The market ptf + state variables describing shifts in the mean-variance frontier EFM 2006/7

  5. Specifics of testing APT • No need to estimate the market ptf • Can be estimated within a subset of the assets • Assume the exact form of APT • In general, approximate APT, which is not testable • The factors and their number are unspecified • Factors can be traded portfolios or not • Factors may explain cross differences in volatility, but have low risk premiums EFM 2006/7

  6. Testing when factors are traded portfolios • With risk-free asset: • Regression of excess asset returns on excess factor returns rt = a + Brf,t + εt, • H0: a=0, F-test • Risk premia: mean excess factor returns • Time-series estimator of variance EFM 2006/7

  7. Testing when factors are traded portfolios(cont.) • Without risk-free asset: need to estimate zero-beta return γ0 • Unconstrained regression of asset returns on factor returns: Rt = a + BRf,t + εt • Constrained regression: Rt = (lN-BlK)γ0 + BRf,t + εt • H0: a=(lN-BlK)γ0, LR test • Risk premia: mean factor returns in excess of γ0 • Variance is adjusted for the estimation error for γ0 EFM 2006/7

  8. Testing when factors are traded portfolios(cont.) • When factor portfolios span the mean-variance frontier: no need to specify zero-beta asset • Regression of asset returns on factor returns: Rt = a + BRf,t + εt • H0: a=0 and BlK=lN • Jensen’s alpha =0 & portfolio weights sum up to 1 • Otherwise, factors do not span the MV frontier of the assets with returns Rt EFM 2006/7

  9. Three approaches to estimate factors • Statistical factors • Extracted from returns • Estimate B and λ at the same time • Macroeconomic factors • Estimate B, then λ • Fundamental factors • Estimate λ for given B (proxied by firm characteristics) EFM 2006/7

  10. Statistical factors:factor analysis Rt- μ = Bft + εt cov(Rt) = B Ω B’ + D • Assuming strict factor structure: • D≡cov(εt) is diagonal • Specification restrictions on factors: • E(ft)=0, Ω≡cov(ft)=I • Estimation: • Estimate B and D by ML • Get ft from the cross-sectional GLS regression of asset returns on B EFM 2006/7

  11. Statistical factors:principal components • Classical approach: • Choose linear combinations of asset returns that maximize explained variance • Each subsequent component is orthogonal to the previous ones • Correspond to the largest eigenvectors of NxN matrix cov(Rt) • Rescaled s.t. weights sum up to 1 EFM 2006/7

  12. Statistical factors:principal components • Connor and Korajczyk (1988): • Take K largest eigenvectors of TxT matrix r’r/N • where r is NxT excess return matrix • As N→∞, KxT matrix of eigenvectors = factor realizations • The factor estimates allow for time-varying risk premiums! • Refinement (like GLS): same for the scaled cross-product matrix r’D-1r/N • where D has variances of the residuals from the first-stage ‘OLS’ on the diagonal, zeros off the diagonal • This increases the efficiency of the estimation EFM 2006/7

  13. Results • 5-6 factors are enough • Based on explicit statistical tests and asset pricing tests • Explain up to 40% of CS variation in stock returns • Better than CAPM • Explain some (January), but not all (size, BE/ME) anomalies EFM 2006/7

  14. Discussion of statistical factors • Missing economic interpretation • The explanatory power out of sample is much lower than in-sample • # factors rises with N • CK fix this problem • Static: slow reaction to the structural changes • Except for CK PCs EFM 2006/7

  15. Macroeconomic factors • Time series to estimate B: Ri,t = ai + b’ift + εi,t • Cross-sectional regressions to estimate ex post risk premia for each t: Ri,t = λ0,t + b'iλK,t + ei,t, • Risk premia: mean and std from the time series of ex post risk premia λt EFM 2006/7

  16. Chen, Roll, and Ross (1986) "Economic forces and the stock market" • Examine the relation between (macro) economic state variables and stock returns • Variables related to CFs / discount rates • Data: • Monthly returns on 20 EW size-sorted portfolios, 1953-1983 EFM 2006/7

  17. Data: macro variables • Industrial production growth: MPt=ln(IPt/IPt-1), YPt=ln(IPt/IPt-12) • Unanticipated inflation: UIt = It – Et-1[It] • Change in expected inflation: DEIt = Et[It+1] – Et-1[It] • Default premium: UPRt = Baat – LGBt • Term premium: UTSt = LGBt – TBt-1 • Real interest rate: RHOt = TBt-1 – It • Market return: EWNYt and VWNYt (NYSE) • Real consumption growth: CG • Change in oil prices: OG EFM 2006/7

  18. Methodology: Fama-MacBeth procedure • Each year, using 20 EW size-sorted portfolios: • Estimate factor loadings B from time-series regression, using previous 5 years Ri,t = ai + b’ift + εi,t • Estimate ex post risk premia from a cross-sectional regression for each of the next 12 months Ri,t = λ0,t + b'iλK,t + ei,t, • Risk premia: mean and std from the time series of ex post risk premia λt EFM 2006/7

  19. Results • Table 4, risk premia • MP: +, insurance against real systematic production risks • UPR: +, hedging against unexpected increases in aggregate risk premium • UTS: - in 1968-77, assets whose prices rise in response to a fall in LR% are more valuable • UI and DEI: - in 1968-77, when they were very volatile • YP, EWNY, VWNY are insignificant EFM 2006/7

  20. Results (cont.) • Table 5, risk premia when market betas are estimated in univariate TS regression • VWNY is significant when alone in CS regression • VWNY is insignificant in the multivariate CS regression • Tables 6 and 7, adding other variables • CG is insignificant • OG: + in 1958-67 EFM 2006/7

  21. Conclusions • Stocks are “exposed to systematic economic news and priced in accordance with their exposures” • Market betas fail to explain CS of stock returns • Though market index is the most significant factor in TS regression • No support for consumption-based pricing EFM 2006/7

  22. Discussionof macroeconomic factors • Strong economic intuition • Static • Slow reaction to the structural changes • Bad predictive performance EFM 2006/7

  23. Fundamental factors • B is proxied by firm characteristics: • Market cap, leverage, E/P, liquidity, etc. • Taken from CAPM violations • Cross-sectional regressions for each t to estimate risk premia: Ri,t = λ0,t + b'iλK,t + ei,t • Alternative: factor-mimicking portfolios • Zero-investment portfolios: long/short position in stocks with high/low value of the attribute EFM 2006/7

  24. Fama and French(1993) "Common risk factors in the returns on stocks and bonds" • Identify risk factors in stock and bond markets • Factors for stocks are size and book-to-market • In contrast to Fama&French (1992): time series tests • Factors for bonds are term structure variables • Links between stock and bond factors EFM 2006/7

  25. Data • All non-financial firms in NYSE, AMEX, and (after 1972) NASDAQ in 1963-1991 • Monthly return data (CRSP) • Annual financial statement data (COMPUSTAT) • Used with a 6m gap • Market index: the CRSP value-wtd portfolio of stocks in the three exchanges EFM 2006/7

  26. Methodology • Stock market factors: • Market: RM-RF • Size: ME • Book-to-market equity: BE/ME • Bond market factors: • TERM = (Return on Long-term Gvt Bonds) – (T-bill rate) • DEF = (Return on Corp Bonds) – (Return on Long-term Gvt Bonds) EFM 2006/7

  27. Constructing factor-mimicking portfolios • In June of each year t, break stocks into: • Two size groups: Small / Big (below/above median) • Three BE/ME groups: Low (bottom 30%) / Medium / High (top 30%) • Compute monthly VW returns of 6 size-BE/ME portfolios for the next 12 months • Factor-mimicking portfolios: zero-investment • Size: SMB = 1/3(SL+SM+SH) – 1/3(BL+BM+BH) • BE/ME: HML = 1/2(BH+SH) – 1/2(BL+SL) EFM 2006/7

  28. The returns to be explained • 25 stock portfolios • In June of each year t, stocks are sorted by size (current ME) and (independently) by BE/ME (as of December of t-1) • Using NYSE quintile breakpoints, all stocks are allocated to one of 5 size portfolios and one of 5 BE/ME portfolios • From July of t to June of t+1, monthly VW returns of 25 size-BE/ME portfolios are computed • 7 bond portfolios • 2 gvt portfolios: 1-5y, 6-10y maturity • 5 corporate bond portfolios: Aaa, Aa, A, Baa, below Baa EFM 2006/7

  29. Time-series tests • Regressions of excess asset returns on factor returns: ri,t = ai + b’irf,t + εi,t • Common variation: slopes and R2 • Pricing: intercepts EFM 2006/7

  30. Results • Table 2: summary statistics • RM-RF, SMB, and HML: high mean and std, (marginally) significant • TERM, DEF: low mean, but high volatility • SMB & HML are almost uncorrelated (-0.08) • RM-RF is positively correlated with SMB (0.32) and negatively with HML (-0.38) EFM 2006/7

  31. Results on common variation • Table 3: explanatory power of bond-market factors • The slopes are higher for stocks, similar to those for long-term bonds • TERM coefficients rise with bond maturity • Small stocks and low-grade bonds are more sensitive to DEF • R2 is higher for bonds EFM 2006/7

  32. Results on common variation(cont.) • Table 4: explanatory power of the market factor • R2 for stocks is much higher, up to 0.9 for small low BE/ME stocks • The slopes for bonds are small, but highly significant, rising with maturity and riskiness • Table 5: explanatory power of SMB and HML • Significant slopes and quite high R2 for stocks • Typically insignificant slopes and zero R2 for bonds EFM 2006/7

  33. Results on common variation(cont.) • Table 6: explanatory power of RM-RF, SMB and HML • Slopes for stocks are highly significant, R2 is typically over 0.9 • Market betas move toward one • The SMB and HML slopes for bonds become significant • Table 7: five-factor regressions • Stocks: stock factors remain significant, but kill significance of bond factors • Bonds: bond factors remain significant, stock factors become much less important • RM-RF help to explain high-grade bonds • SMB and HML help to explain low-grade bonds EFM 2006/7

  34. Results on common variation(cont.) • Orthogonalization of the market factor: RM-RF=0.5+0.44SMB-0.63HML+0.81TERM+0.79DEF+e • All coefficients are significant, R2=0.38 • The market factor captures common variation in stock and bond markets! • Orthogonalized market factor: RMO = const + error • Table 8: five-factor regressions with RMO • Stocks: bond factors become highly significant EFM 2006/7

  35. Results on pricing • Table 9a, stocks • TERM, DEF: positive intercepts • RM-RF: size effect • SMB, HML: big positive intercepts • RM-RF, SMB, HML: most intercepts are 0 • Adding bond factors does not improve EFM 2006/7

  36. Results on pricing(cont.) • Table 9b, bonds • TERM, DEF: positive intercepts for gvt bonds • RM-RF or SMB with HML make intercepts insignificant • Increased precision due to TERM and DEF explains positive intercepts in a five-factor model • Table 9c, F-test • Joint test for zero intercepts rejects the null for all models • The best model for stocks is a model with three stock factors EFM 2006/7

  37. Diagnostics • Time series regressions of residuals from the five-factor model on D/P, default spread, term spread, and short-term interest rates • No evidence of predictability! • Table 10, time series regressions of residuals on January dummy • January seasonals are weak, mostly for small and high BE/ME stocks • Except for TERM, there are January seasonals in risk factors, esp. in SMB and HML EFM 2006/7

  38. Split-sample tests • Each of the size-BE/ME portfolios is split into two halves • One is used to form factors • Another is used as dependent variables in regressions • Similar results EFM 2006/7

  39. Other sets of portfolios • Portfolios formed on E/P • Zero intercepts • Portfolios formed on D/P • The only unexplained portfolio: D=0, a=-0.23 EFM 2006/7

  40. Conclusions • There is an overlap between processes in stock and bond markets • Bond market factors capture common variation in stock and bond returns, though explain almost no average excess stock returns • Three-factor model with the market, size, and book-to-market factors explains well stock returns • SMB and HML explain the cross differences • RM-RF explains why stock returns are on average above the T-bill rate • Two bond factors explain well variation in bond returns • SMB and HML help to explain variation of low-grade bonds EFM 2006/7

  41. Fama and French(1995) "Size and book-to-market factors in earnings and returns" • There are size and book-to-market factors in earnings which proxy for relative distress • Strong firms with persistently high earnings have low BE/ME • Small stocks tend to be less profitable • There is some relation between common factors in earnings and return variation EFM 2006/7

  42. Fama and French(1996) "Multifactor explanations of asset pricing anomalies" • Run time-series regressions for decile portfolios based on sorting by E/P, C/P, sales, past returns • The three-factor model explains all anomalies but one-year momentum effect • Interpretation of the three-factor model in terms of the underlying portfolios M, S, B, H, and L: spanning tests • M and B are highly correlated (0.99) • Excess returns of any three of M, S, H, and L explain the fourth • Different triplets of the excess returns for M, S, H, and L provide similar results in explaining stock returns • This is taken as evidence of multifactor ICAPM or APT EFM 2006/7

  43. Discussionof fundamental factors • High predictive power • Dynamic • Though: data-intensive • Widely applied: • Portfolio selection and risk management • Performance evaluation • Measuring abnormal returns in event studies • Estimating the cost of capital EFM 2006/7

More Related